firm theory - 1.5 cost minimization with CD production function (three inputs).ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import necessary packages."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# import all functions from python labrary: sympy\n",
    "from sympy import *\n",
    "# import python library: matplotlib (for nice plots)\n",
    "import matplotlib\n",
    "# set sympy output to print pretty math expressions\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Declare variables we are going to use as 'symbols'. We will add more along the way, if necessary."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "### NOTE: it is important here to further declare these variables to be nonzero and positive (otherwise sympy cannot find solution for the FOCs)\n",
    "#    Thanks to Abhi Agarwal for discovering this trick\n",
    "L, K, M, w, r, m, qbar = symbols('L K M w r m \\overline{q}', nonzero=True, positive=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solve for cost-minimizing input bundle analytically"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the optimization program"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The production manager's optimization problem with CD production function and three inputs, labor, capital and material, is\n",
    "\\begin{align}\n",
    "    & \\min_{L,K,M} \\quad wL + rK + mM \\\\\n",
    "    \\text{s.t.}\\quad & L^{\\alpha} K^{\\beta} M^{\\gamma} = \\overline{q}\n",
    "\\end{align}\n",
    "where $m$ is the constant unit cost of material (M)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Lagrangian is\n",
    "\\begin{align}\n",
    "    \\mathcal{L} = wL + rK + mM + \\lambda ( \\overline{q} - L^{\\alpha} K^{\\beta} M^{\\gamma})\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The FOCs are\n",
    "\\begin{align}\n",
    "    w - \\lambda \\alpha L^{\\alpha-1} K^{\\beta} M^{\\gamma} &= 0  \\tag{1} \\\\\n",
    "    r - \\lambda \\beta L^{\\alpha} K^{\\beta-1} M^{\\gamma} &= 0 \\tag{2} \\\\\n",
    "    m - \\lambda \\gamma L^{\\alpha} K^{\\beta} M^{\\gamma-1} &= 0 \\tag{3} \\\\\n",
    "    \\overline{q} - L^{\\alpha} K^{\\beta} M^{\\gamma} &= 0  \\tag{4} \n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or, equivalently,\n",
    "\\begin{align}\n",
    "    w &= \\lambda \\alpha L^{\\alpha-1} K^{\\beta} M^{\\gamma}   \\tag{1} \\\\\n",
    "    r &= \\lambda \\beta L^{\\alpha} K^{\\beta-1} M^{\\gamma}  \\tag{2} \\\\\n",
    "    m &= \\lambda \\gamma L^{\\alpha} K^{\\beta} M^{\\gamma-1}  \\tag{3} \\\\\n",
    "    \\overline{q} &= L^{\\alpha} K^{\\beta} M^{\\gamma}   \\tag{4} \n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dividing (1) by (2), we have\n",
    "\\begin{align}\n",
    "    \\frac{w}{r} &= \\frac{\\alpha}{\\beta}\\frac{K}{L}\n",
    "\\end{align}\n",
    "or \n",
    "\\begin{align}\n",
    "    L &= \\frac{\\alpha}{\\beta} \\frac{r}{w} K \\tag{5}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dividing (3) by (2), we have\n",
    "\\begin{align}\n",
    "    \\frac{m}{r} &= \\frac{\\gamma}{\\beta}\\frac{K}{M}\n",
    "\\end{align}\n",
    "or \n",
    "\\begin{align}\n",
    "    M &= \\frac{\\gamma}{\\beta} \\frac{r}{m} K \\tag{6}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that both (5) and (6) are functions of K, we plug them into (4)\n",
    "\\begin{align}\n",
    "    \\overline{q} &= \\left( \\frac{\\alpha}{\\beta} \\frac{r}{w} K \\right)^{\\alpha} K^{\\beta} \\left( \\frac{\\gamma}{\\beta} \\frac{r}{m} K  \\right)^{\\gamma} \\\\\n",
    "    \\overline{q} &= \\left( \\frac{\\alpha}{\\beta} \\frac{r}{w} \\right)^{\\alpha} \\left( \\frac{\\gamma}{\\beta} \\frac{r}{m}  \\right)^{\\gamma} K^{\\alpha + \\beta + \\gamma}  \\\\\n",
    "    K^{\\alpha + \\beta + \\gamma} &= \\left( \\frac{\\beta}{\\alpha} \\frac{w}{r} \\right)^{\\alpha} \\left( \\frac{\\beta}{\\gamma} \\frac{m}{r}  \\right)^{\\gamma} \\overline{q}  \\\\\n",
    "    K^{\\alpha + \\beta + \\gamma} &= \\left( \\frac{\\beta}{r} \\frac{w}{\\alpha} \\right)^{\\alpha} \\left( \\frac{\\beta}{r} \\frac{m}{\\gamma}  \\right)^{\\gamma} \\overline{q}  \\\\\n",
    "    K^{\\alpha + \\beta + \\gamma} &= \\left( \\frac{\\beta}{r} \\right)^{\\alpha + \\gamma} \\left( \\frac{w}{\\alpha} \\right)^{\\alpha} \\left( \\frac{m}{\\gamma}  \\right)^{\\gamma} \\overline{q}  \\\\\n",
    "    K^{\\alpha + \\beta + \\gamma} &= \\left( \\frac{\\beta}{r} \\frac{w}{\\alpha} \\right)^{\\alpha} \\left( \\frac{\\beta}{r} \\frac{m}{\\gamma}  \\right)^{\\gamma} \\overline{q}  \\\\\n",
    "    K^{\\alpha + \\beta + \\gamma} &= \\left( \\frac{\\beta}{r} \\right)^{\\alpha + \\gamma} \\left( \\frac{w}{\\alpha} \\right)^{\\alpha} \\left( \\frac{m}{\\gamma}  \\right)^{\\gamma} \\overline{q}  \\\\\n",
    "    K &= \\left[ \\left( \\frac{\\beta}{r} \\right)^{\\alpha + \\gamma} \\left( \\frac{w}{\\alpha} \\right)^{\\alpha} \\left( \\frac{m}{\\gamma}  \\right)^{\\gamma} \\overline{q} \\right]^{\\frac{1}{\\alpha + \\beta + \\gamma} } \n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "which is the optimal demand for $K$, and it can rewritten as \n",
    "\\begin{align} \n",
    "    K^* &= \\left( \\frac{\\beta}{r} \\right)^{ \\frac{\\alpha + \\gamma}{\\alpha + \\beta + \\gamma} } \\left( \\frac{w}{\\alpha} \\right)^{ \\frac{\\alpha}{\\alpha + \\beta + \\gamma} } \\left( \\frac{m}{\\gamma}  \\right)^{ \\frac{\\gamma}{\\alpha + \\beta + \\gamma} } \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} }   \\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plugging $K^*$ into (5), we can find the optimal demand for labor\n",
    "\\begin{align}\n",
    "    L^* &= \\frac{\\alpha}{\\beta} \\frac{r}{w} K^* \\\\\n",
    "        &= \\frac{\\alpha}{w} \\frac{r}{\\beta} K^* \\\\\n",
    "        &= \\left( \\frac{w}{\\alpha} \\right)^{-1} \\left( \\frac{\\beta}{r}  \\right)^{-1} K^* \\\\\n",
    "        &= \\left( \\frac{w}{\\alpha} \\right)^{-1} \\left( \\frac{\\beta}{r}  \\right)^{-1} \\left( \\frac{\\beta}{r} \\right)^{ \\frac{\\alpha + \\gamma}{\\alpha + \\beta + \\gamma} } \\left( \\frac{w}{\\alpha} \\right)^{ \\frac{\\alpha}{\\alpha + \\beta + \\gamma} } \\left( \\frac{m}{\\gamma}  \\right)^{ \\frac{\\gamma}{\\alpha + \\beta + \\gamma} } \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} } \\\\\n",
    "        &= \\left( \\frac{w}{\\alpha} \\right)^{- \\frac{\\alpha + \\beta + \\gamma}{\\alpha + \\beta + \\gamma} } \\left( \\frac{\\beta}{r}  \\right)^{- \\frac{\\alpha + \\beta + \\gamma}{\\alpha + \\beta + \\gamma}} \\left( \\frac{\\beta}{r} \\right)^{ \\frac{\\alpha + \\gamma}{\\alpha + \\beta + \\gamma} } \\left( \\frac{w}{\\alpha} \\right)^{ \\frac{\\alpha}{\\alpha + \\beta + \\gamma} } \\left( \\frac{m}{\\gamma}  \\right)^{ \\frac{\\gamma}{\\alpha + \\beta + \\gamma} } \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} } \\\\\n",
    "        &= \\left( \\frac{\\alpha}{w} \\right)^{ \\frac{\\beta + \\gamma}{\\alpha + \\beta + \\gamma} } \\left( \\frac{r}{\\beta}  \\right)^{ \\frac{ \\beta }{\\alpha + \\beta + \\gamma}}  \\left( \\frac{m}{\\gamma}  \\right)^{ \\frac{\\gamma}{\\alpha + \\beta + \\gamma} } \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} } \\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plugging $K^*$ into (6), we can find the optimal demand for material\n",
    "\\begin{align}\n",
    "    M^* &= \\frac{\\gamma}{\\beta} \\frac{r}{m} K^* \\\\\n",
    "        &= \\frac{r}{\\beta} \\frac{\\gamma}{m} K^* \\\\\n",
    "        &= \\left( \\frac{\\beta}{r}  \\right)^{-1} \\left( \\frac{m}{\\gamma} \\right)^{-1} K^* \\\\\n",
    "        &= \\left( \\frac{\\beta}{r}  \\right)^{-1} \\left( \\frac{m}{\\gamma} \\right)^{-1} \\left( \\frac{\\beta}{r} \\right)^{ \\frac{\\alpha + \\gamma}{\\alpha + \\beta + \\gamma} } \\left( \\frac{w}{\\alpha} \\right)^{ \\frac{\\alpha}{\\alpha + \\beta + \\gamma} } \\left( \\frac{m}{\\gamma}  \\right)^{ \\frac{\\gamma}{\\alpha + \\beta + \\gamma} } \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} } \\\\\n",
    "        &= \\left( \\frac{\\beta}{r}  \\right)^{- \\frac{\\alpha + \\beta + \\gamma}{\\alpha + \\beta + \\gamma} } \\left( \\frac{m}{\\gamma}  \\right)^{- \\frac{\\alpha + \\beta + \\gamma}{\\alpha + \\beta + \\gamma}} \\left( \\frac{\\beta}{r} \\right)^{ \\frac{\\alpha + \\gamma}{\\alpha + \\beta + \\gamma} } \\left( \\frac{w}{\\alpha} \\right)^{ \\frac{\\alpha}{\\alpha + \\beta + \\gamma} } \\left( \\frac{m}{\\gamma}  \\right)^{ \\frac{\\gamma}{\\alpha + \\beta + \\gamma} } \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} } \\\\\n",
    "        &= \\left( \\frac{\\gamma}{m}  \\right)^{ \\frac{\\alpha + \\beta}{\\alpha + \\beta + \\gamma} } \\left( \\frac{w}{\\alpha} \\right)^{ \\frac{\\alpha}{\\alpha + \\beta + \\gamma} } \\left( \\frac{r}{\\beta}  \\right)^{ \\frac{ \\beta }{\\alpha + \\beta + \\gamma}}   \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} } \\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plugging in the optimal demands into cost, we have the cost function\n",
    "\\begin{align}\n",
    "    C &= wL^* + rK^* + mM^* \\\\\n",
    "      &= w \\left( \\frac{\\alpha}{w} \\right)^{ \\frac{\\beta + \\gamma}{\\alpha + \\beta + \\gamma} } \\left( \\frac{r}{\\beta}  \\right)^{ \\frac{ \\beta }{\\alpha + \\beta + \\gamma}}  \\left( \\frac{m}{\\gamma}  \\right)^{ \\frac{\\gamma}{\\alpha + \\beta + \\gamma} } \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} }  \n",
    "      \\\\\n",
    "      & \\quad +  r \\left( \\frac{\\beta}{r} \\right)^{ \\frac{\\alpha + \\gamma}{\\alpha + \\beta + \\gamma} } \\left( \\frac{w}{\\alpha} \\right)^{ \\frac{\\alpha}{\\alpha + \\beta + \\gamma} } \\left( \\frac{m}{\\gamma}  \\right)^{ \\frac{\\gamma}{\\alpha + \\beta + \\gamma} } \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} }  \n",
    "      \\\\\n",
    "      & \\quad +  m \\left( \\frac{\\gamma}{m}  \\right)^{ \\frac{\\alpha + \\beta}{\\alpha + \\beta + \\gamma} } \\left( \\frac{w}{\\alpha} \\right)^{ \\frac{\\alpha}{\\alpha + \\beta + \\gamma} } \\left( \\frac{r}{\\beta}  \\right)^{ \\frac{ \\beta }{\\alpha + \\beta + \\gamma}}   \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} } \n",
    "      \\\\\n",
    "    &= w \\left( \\frac{\\alpha}{w} \\right)^{ \\frac{\\alpha + \\beta + \\gamma}{\\alpha + \\beta + \\gamma} } \\left( \\frac{\\alpha}{w} \\right)^{- \\frac{\\alpha}{\\alpha + \\beta + \\gamma} } \\left( \\frac{r}{\\beta}  \\right)^{ \\frac{ \\beta }{\\alpha + \\beta + \\gamma}}  \\left( \\frac{m}{\\gamma}  \\right)^{ \\frac{\\gamma}{\\alpha + \\beta + \\gamma} } \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} }   \n",
    "      \\\\\n",
    "      & \\quad +  r \\left( \\frac{\\beta}{r} \\right)^{ \\frac{\\alpha + \\beta + \\gamma}{\\alpha + \\beta + \\gamma} } \\left( \\frac{\\beta}{r} \\right)^{ -\\frac{\\beta}{\\alpha + \\beta + \\gamma} } \\left( \\frac{w}{\\alpha} \\right)^{ \\frac{\\alpha}{\\alpha + \\beta + \\gamma} } \\left( \\frac{m}{\\gamma}  \\right)^{ \\frac{\\gamma}{\\alpha + \\beta + \\gamma} } \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} } \n",
    "      \\\\\n",
    "      & \\quad + m \\left( \\frac{\\gamma}{m}  \\right)^{ \\frac{\\alpha + \\beta + \\gamma}{\\alpha + \\beta + \\gamma} } \\left( \\frac{\\gamma}{m}  \\right)^{ -\\frac{\\gamma}{\\alpha + \\beta + \\gamma} } \\left( \\frac{w}{\\alpha} \\right)^{ \\frac{\\alpha}{\\alpha + \\beta + \\gamma} } \\left( \\frac{r}{\\beta}  \\right)^{ \\frac{ \\beta }{\\alpha + \\beta + \\gamma}}   \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} }  \n",
    "      \\\\\n",
    "    &= w \\left( \\frac{\\alpha}{w} \\right) \\left( \\frac{w}{\\alpha} \\right)^{ \\frac{\\alpha}{\\alpha + \\beta + \\gamma} } \\left( \\frac{r}{\\beta}  \\right)^{ \\frac{ \\beta }{\\alpha + \\beta + \\gamma}}  \\left( \\frac{m}{\\gamma}  \\right)^{ \\frac{\\gamma}{\\alpha + \\beta + \\gamma} } \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} }  \n",
    "      \\\\\n",
    "      & \\quad +  r \\left( \\frac{\\beta}{r} \\right) \\left( \\frac{r}{\\beta} \\right)^{ \\frac{\\beta}{\\alpha + \\beta + \\gamma} } \\left( \\frac{w}{\\alpha} \\right)^{ \\frac{\\alpha}{\\alpha + \\beta + \\gamma} } \\left( \\frac{m}{\\gamma}  \\right)^{ \\frac{\\gamma}{\\alpha + \\beta + \\gamma} } \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} }  \n",
    "      \\\\\n",
    "      & \\quad +  m \\left( \\frac{\\gamma}{m}  \\right) \\left( \\frac{m}{\\gamma}  \\right)^{ \\frac{\\gamma}{\\alpha + \\beta + \\gamma} } \\left( \\frac{w}{\\alpha} \\right)^{ \\frac{\\alpha}{\\alpha + \\beta + \\gamma} } \\left( \\frac{r}{\\beta}  \\right)^{ \\frac{ \\beta }{\\alpha + \\beta + \\gamma}}   \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} } \n",
    "    \\\\\n",
    "    &= (\\alpha + \\beta + \\gamma) \\left( \\frac{w}{\\alpha} \\right)^{ \\frac{\\alpha}{\\alpha + \\beta + \\gamma} } \\left( \\frac{r}{\\beta}  \\right)^{ \\frac{ \\beta }{\\alpha + \\beta + \\gamma}}  \\left( \\frac{m}{\\gamma}  \\right)^{ \\frac{\\gamma}{\\alpha + \\beta + \\gamma} } \\overline{q}^{ \\frac{1}{\\alpha + \\beta + \\gamma} }\n",
    "\\end{align}\n",
    "\n",
    "Note that the cost function is a function of input prices, output level and productivity parameters."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sympy solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the utility function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle K^{\\beta} L^{\\alpha} M^{\\gamma}$"
      ],
      "text/plain": [
       " β  α  γ\n",
       "K ⋅L ⋅M "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "### Note that it is important here to further declare these variables to be nonzero and positive\n",
    "#    Thanks to Abhi Agarwal for discovering this trick\n",
    "alpha, beta, gamma = symbols('alpha, beta, gamma',nonzero=True, positive=True)\n",
    "\n",
    "\n",
    "# Note: in sympy syntax: use double star ** for power\n",
    "q = L**(alpha) * K**(beta) * M**(gamma)\n",
    "# show the defined function\n",
    "q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the lagrangian"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle K r + L w + M m + \\lambda \\left(- K^{\\beta} L^{\\alpha} M^{\\gamma} + \\overline{q}\\right)$"
      ],
      "text/plain": [
       "                    ⎛   β  α  γ               ⎞\n",
       "K⋅r + L⋅w + M⋅m + λ⋅⎝- K ⋅L ⋅M  + \\overline{q}⎠"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lam = symbols('lambda',nonzero=True, positive=True)\n",
    "lagr = w*L + r*K + m*M + lam *(qbar - q)\n",
    "lagr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solve the first-order conditions. \n",
    "\n",
    "Note the syntax of the solve function, especially the use of the brackets:\n",
    "solve( (eq1, eq2, eq3), var1, var2, var3)\n",
    "\n",
    "The equations are by default treated as equal to 0 by the solver."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left( \\frac{\\overline{q} \\alpha \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} \\alpha^{- \\alpha} \\beta^{- \\beta} \\gamma^{- \\gamma} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}}{w}, \\  \\left(\\overline{q} \\left(m \\left(\\alpha^{\\alpha} \\beta^{\\beta} \\gamma^{\\gamma}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{\\gamma} \\left(w \\left(\\alpha^{\\alpha} \\beta^{\\beta} \\gamma^{\\gamma}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{\\alpha} \\left(\\overline{q} \\alpha \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{- \\alpha} \\left(\\overline{q} \\gamma \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{- \\gamma}\\right)^{\\frac{1}{\\beta}}, \\  \\frac{\\overline{q} \\gamma \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} \\alpha^{- \\alpha} \\beta^{- \\beta} \\gamma^{- \\gamma} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}}{m}, \\  \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} \\alpha^{- \\alpha} \\beta^{- \\beta} \\gamma^{- \\gamma} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)\\right]$"
      ],
      "text/plain": [
       "⎡⎛                                                                            \n",
       "⎢⎜                                                                    1       \n",
       "⎢⎜                                                                ─────────   \n",
       "⎢⎜                                                                α + β + γ   \n",
       "⎢⎜               ⎛            -α - β - γ + 1  -α  -β  -γ  γ  β  α⎞            \n",
       "⎢⎜\\overline{q}⋅α⋅⎝\\overline{q}              ⋅α  ⋅β  ⋅γ  ⋅m ⋅r ⋅w ⎠           b\n",
       "⎢⎜─────────────────────────────────────────────────────────────────────────,  \n",
       "⎣⎝                                    w                                       \n",
       "\n",
       "        ______________________________________________________________________\n",
       "       ╱                                     γ                        α       \n",
       "      ╱               ⎛                1    ⎞  ⎛                1    ⎞  ⎛     \n",
       "     ╱                ⎜            ─────────⎟  ⎜            ─────────⎟  ⎜     \n",
       "    ╱                 ⎜            α + β + γ⎟  ⎜            α + β + γ⎟  ⎜     \n",
       "eta╱                  ⎜  ⎛ α  β  γ⎞         ⎟  ⎜  ⎛ α  β  γ⎞         ⎟  ⎜     \n",
       " ╲╱      \\overline{q}⋅⎝m⋅⎝α ⋅β ⋅γ ⎠         ⎠ ⋅⎝w⋅⎝α ⋅β ⋅γ ⎠         ⎠ ⋅⎝\\over\n",
       "                                                                              \n",
       "\n",
       "______________________________________________________________________________\n",
       "                                                         -α                   \n",
       "                                                   1    ⎞   ⎛                 \n",
       "                                               ─────────⎟   ⎜                 \n",
       "                                               α + β + γ⎟   ⎜                 \n",
       "          ⎛            -α - β - γ + 1  γ  β  α⎞         ⎟   ⎜               ⎛ \n",
       "line{q}⋅α⋅⎝\\overline{q}              ⋅m ⋅r ⋅w ⎠         ⎠  ⋅⎝\\overline{q}⋅γ⋅⎝\\\n",
       "                                                                              \n",
       "\n",
       "________________________________________________                              \n",
       "                                             -γ                               \n",
       "                                       1    ⎞                                 \n",
       "                                   ─────────⎟                                 \n",
       "                                   α + β + γ⎟                    ⎛            \n",
       "           -α - β - γ + 1  γ  β  α⎞         ⎟     \\overline{q}⋅γ⋅⎝\\overline{q}\n",
       "overline{q}              ⋅m ⋅r ⋅w ⎠         ⎠   , ────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                        1                                     \n",
       "                                    ─────────                                 \n",
       "                                    α + β + γ                                 \n",
       "-α - β - γ + 1  -α  -β  -γ  γ  β  α⎞                                          \n",
       "              ⋅α  ⋅β  ⋅γ  ⋅m ⋅r ⋅w ⎠           ⎛            -α - β - γ + 1  -α\n",
       "─────────────────────────────────────────────, ⎝\\overline{q}              ⋅α  \n",
       "        m                                                                     \n",
       "\n",
       "                           ⎞⎤\n",
       "                           ⎟⎥\n",
       "                      1    ⎟⎥\n",
       "                  ─────────⎟⎥\n",
       "                  α + β + γ⎟⎥\n",
       "  -β  -γ  γ  β  α⎞         ⎟⎥\n",
       "⋅β  ⋅γ  ⋅m ⋅r ⋅w ⎠         ⎟⎥\n",
       "                           ⎠⎦"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "foc_sol = solve( ( diff(lagr,L), diff(lagr,K), diff(lagr,M), diff(lagr,lam) ), [L,K,M,lam])\n",
    "foc_sol"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The solution is a list with one element, because our solution is unique."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{\\overline{q} \\alpha \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} \\alpha^{- \\alpha} \\beta^{- \\beta} \\gamma^{- \\gamma} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}}{w}, \\  \\left(\\overline{q} \\left(m \\left(\\alpha^{\\alpha} \\beta^{\\beta} \\gamma^{\\gamma}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{\\gamma} \\left(w \\left(\\alpha^{\\alpha} \\beta^{\\beta} \\gamma^{\\gamma}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{\\alpha} \\left(\\overline{q} \\alpha \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{- \\alpha} \\left(\\overline{q} \\gamma \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{- \\gamma}\\right)^{\\frac{1}{\\beta}}, \\  \\frac{\\overline{q} \\gamma \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} \\alpha^{- \\alpha} \\beta^{- \\beta} \\gamma^{- \\gamma} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}}{m}, \\  \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} \\alpha^{- \\alpha} \\beta^{- \\beta} \\gamma^{- \\gamma} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)$"
      ],
      "text/plain": [
       "⎛                                                                             \n",
       "⎜                                                                    1        \n",
       "⎜                                                                ─────────    \n",
       "⎜                                                                α + β + γ    \n",
       "⎜               ⎛            -α - β - γ + 1  -α  -β  -γ  γ  β  α⎞             \n",
       "⎜\\overline{q}⋅α⋅⎝\\overline{q}              ⋅α  ⋅β  ⋅γ  ⋅m ⋅r ⋅w ⎠           be\n",
       "⎜─────────────────────────────────────────────────────────────────────────,   \n",
       "⎝                                    w                                        \n",
       "\n",
       "       _______________________________________________________________________\n",
       "      ╱                                     γ                        α        \n",
       "     ╱               ⎛                1    ⎞  ⎛                1    ⎞  ⎛      \n",
       "    ╱                ⎜            ─────────⎟  ⎜            ─────────⎟  ⎜      \n",
       "   ╱                 ⎜            α + β + γ⎟  ⎜            α + β + γ⎟  ⎜      \n",
       "ta╱                  ⎜  ⎛ α  β  γ⎞         ⎟  ⎜  ⎛ α  β  γ⎞         ⎟  ⎜      \n",
       "╲╱      \\overline{q}⋅⎝m⋅⎝α ⋅β ⋅γ ⎠         ⎠ ⋅⎝w⋅⎝α ⋅β ⋅γ ⎠         ⎠ ⋅⎝\\overl\n",
       "                                                                              \n",
       "\n",
       "______________________________________________________________________________\n",
       "                                                        -α                    \n",
       "                                                  1    ⎞   ⎛                  \n",
       "                                              ─────────⎟   ⎜                  \n",
       "                                              α + β + γ⎟   ⎜                  \n",
       "         ⎛            -α - β - γ + 1  γ  β  α⎞         ⎟   ⎜               ⎛  \n",
       "ine{q}⋅α⋅⎝\\overline{q}              ⋅m ⋅r ⋅w ⎠         ⎠  ⋅⎝\\overline{q}⋅γ⋅⎝\\o\n",
       "                                                                              \n",
       "\n",
       "_______________________________________________                               \n",
       "                                            -γ                                \n",
       "                                      1    ⎞                                  \n",
       "                                  ─────────⎟                                  \n",
       "                                  α + β + γ⎟                    ⎛            -\n",
       "          -α - β - γ + 1  γ  β  α⎞         ⎟     \\overline{q}⋅γ⋅⎝\\overline{q} \n",
       "verline{q}              ⋅m ⋅r ⋅w ⎠         ⎠   , ─────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                       1                                      \n",
       "                                   ─────────                                  \n",
       "                                   α + β + γ                                  \n",
       "α - β - γ + 1  -α  -β  -γ  γ  β  α⎞                                           \n",
       "             ⋅α  ⋅β  ⋅γ  ⋅m ⋅r ⋅w ⎠           ⎛            -α - β - γ + 1  -α \n",
       "────────────────────────────────────────────, ⎝\\overline{q}              ⋅α  ⋅\n",
       "       m                                                                      \n",
       "\n",
       "                          ⎞\n",
       "                          ⎟\n",
       "                     1    ⎟\n",
       "                 ─────────⎟\n",
       "                 α + β + γ⎟\n",
       " -β  -γ  γ  β  α⎞         ⎟\n",
       "β  ⋅γ  ⋅m ⋅r ⋅w ⎠         ⎟\n",
       "                          ⎠"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "foc_sol[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To access the first element in our solution, which is a list by itself, we use the list index again."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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9Vlbxu6QHfhTmK4Zf5PmYlDlD4GgQMMNwNKK2jnoEXvmvcb1XPP107QulnUVI8Y1f4nw61JwhcDQImGE4GlFbR7XyR8nzTXMcX+VKP7HJh6IwGM7JgPBtZr5Lnpa7KmB/DYGdImCGYaeCtW5lEbiUsn/jcx7peR5KyWjcVZgjI4xB7D6qjhmGGBEL7x4BMwy7F7F1MCAgBc/REN8YZ2eAEQhGgmTSDirDDiF2rk6cYGFDYO8ImGHYu4StfzkE+I7v52AofAGOlj7Ghf0uojhaivMsbAjsGQEzDHuWrvWtDoEzZaTHQ6Slu4XHMh7p0VIdTUs3BHaDgBmG3YjSOtIDAQwAhsA5f7R0W5E0zXYLHiN7HBcC9gW345K39dYjIGMQfq/AHcI3ee4b3spfyN+R/6LdQnwHoSRzhsBxIPB/F3bKFAZyZvUAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\overline{q} \\alpha \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} \\alpha^{- \\alpha} \\beta^{- \\beta} \\gamma^{- \\gamma} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}}{w}$"
      ],
      "text/plain": [
       "                                                                    1    \n",
       "                                                                ─────────\n",
       "                                                                α + β + γ\n",
       "               ⎛            -α - β - γ + 1  -α  -β  -γ  γ  β  α⎞         \n",
       "\\overline{q}⋅α⋅⎝\\overline{q}              ⋅α  ⋅β  ⋅γ  ⋅m ⋅r ⋅w ⎠         \n",
       "─────────────────────────────────────────────────────────────────────────\n",
       "                                    w                                    "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Lstar = foc_sol[0][0].simplify()\n",
    "Lstar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left(\\overline{q} \\left(m \\left(\\alpha^{\\alpha} \\beta^{\\beta} \\gamma^{\\gamma}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{\\gamma} \\left(w \\left(\\alpha^{\\alpha} \\beta^{\\beta} \\gamma^{\\gamma}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{\\alpha} \\left(\\overline{q} \\alpha \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{- \\alpha} \\left(\\overline{q} \\gamma \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{- \\gamma}\\right)^{\\frac{1}{\\beta}}$"
      ],
      "text/plain": [
       "         _____________________________________________________________________\n",
       "        ╱                                     γ                        α      \n",
       "       ╱               ⎛                1    ⎞  ⎛                1    ⎞  ⎛    \n",
       "      ╱                ⎜            ─────────⎟  ⎜            ─────────⎟  ⎜    \n",
       "     ╱                 ⎜            α + β + γ⎟  ⎜            α + β + γ⎟  ⎜    \n",
       "beta╱                  ⎜  ⎛ α  β  γ⎞         ⎟  ⎜  ⎛ α  β  γ⎞         ⎟  ⎜    \n",
       "  ╲╱      \\overline{q}⋅⎝m⋅⎝α ⋅β ⋅γ ⎠         ⎠ ⋅⎝w⋅⎝α ⋅β ⋅γ ⎠         ⎠ ⋅⎝\\ove\n",
       "\n",
       "______________________________________________________________________________\n",
       "                                                          -α                  \n",
       "                                                    1    ⎞   ⎛                \n",
       "                                                ─────────⎟   ⎜                \n",
       "                                                α + β + γ⎟   ⎜                \n",
       "           ⎛            -α - β - γ + 1  γ  β  α⎞         ⎟   ⎜               ⎛\n",
       "rline{q}⋅α⋅⎝\\overline{q}              ⋅m ⋅r ⋅w ⎠         ⎠  ⋅⎝\\overline{q}⋅γ⋅⎝\n",
       "\n",
       "_________________________________________________\n",
       "                                              -γ \n",
       "                                        1    ⎞   \n",
       "                                    ─────────⎟   \n",
       "                                    α + β + γ⎟   \n",
       "            -α - β - γ + 1  γ  β  α⎞         ⎟   \n",
       "\\overline{q}              ⋅m ⋅r ⋅w ⎠         ⎠   "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Kstar = foc_sol[0][1].simplify()\n",
    "Kstar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\overline{q} \\gamma \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} \\alpha^{- \\alpha} \\beta^{- \\beta} \\gamma^{- \\gamma} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}}{m}$"
      ],
      "text/plain": [
       "                                                                    1    \n",
       "                                                                ─────────\n",
       "                                                                α + β + γ\n",
       "               ⎛            -α - β - γ + 1  -α  -β  -γ  γ  β  α⎞         \n",
       "\\overline{q}⋅γ⋅⎝\\overline{q}              ⋅α  ⋅β  ⋅γ  ⋅m ⋅r ⋅w ⎠         \n",
       "─────────────────────────────────────────────────────────────────────────\n",
       "                                    m                                    "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Mstar = foc_sol[0][2].simplify()\n",
    "Mstar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Substitute optimal input demands into the cost, to calculate the minimum cost level necessary at the given exogenous variables. \n",
    "\n",
    "The resulting function is our **cost function**, $C(w,r,\\overline{q})$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left(\\left(\\overline{q} \\alpha \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{- \\alpha} \\left(\\overline{q} \\gamma \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{- \\gamma}\\right)^{\\frac{1}{\\beta}} \\left(\\alpha^{- \\alpha} \\beta^{- \\beta} \\gamma^{- \\gamma}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}} \\left(\\overline{q} \\left(\\left(\\overline{q} \\alpha \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{\\alpha} \\left(\\overline{q} \\gamma \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{\\gamma}\\right)^{\\frac{1}{\\beta}} \\left(\\overline{q}^{- \\alpha - \\beta - \\gamma + 1} m^{\\gamma} r^{\\beta} w^{\\alpha}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}} \\left(\\alpha + \\gamma\\right) + r \\left(\\overline{q} \\left(m \\left(\\alpha^{\\alpha} \\beta^{\\beta} \\gamma^{\\gamma}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{\\gamma} \\left(w \\left(\\alpha^{\\alpha} \\beta^{\\beta} \\gamma^{\\gamma}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)^{\\alpha}\\right)^{\\frac{1}{\\beta}} \\left(\\alpha^{\\alpha} \\beta^{\\beta} \\gamma^{\\gamma}\\right)^{\\frac{1}{\\alpha + \\beta + \\gamma}}\\right)$"
      ],
      "text/plain": [
       "         _____________________________________________________________________\n",
       "        ╱                                                                -α   \n",
       "       ╱  ⎛                                                        1    ⎞   ⎛ \n",
       "      ╱   ⎜                                                    ─────────⎟   ⎜ \n",
       "     ╱    ⎜                                                    α + β + γ⎟   ⎜ \n",
       "beta╱     ⎜               ⎛            -α - β - γ + 1  γ  β  α⎞         ⎟   ⎜ \n",
       "  ╲╱      ⎝\\overline{q}⋅α⋅⎝\\overline{q}              ⋅m ⋅r ⋅w ⎠         ⎠  ⋅⎝\\\n",
       "\n",
       "________________________________________________________________              \n",
       "                                                             -γ               \n",
       "                                                       1    ⎞                 \n",
       "                                                   ─────────⎟                 \n",
       "                                                   α + β + γ⎟                 \n",
       "              ⎛            -α - β - γ + 1  γ  β  α⎞         ⎟    ⎛ -α  -β  -γ⎞\n",
       "overline{q}⋅γ⋅⎝\\overline{q}              ⋅m ⋅r ⋅w ⎠         ⎠   ⋅⎝α  ⋅β  ⋅γ  ⎠\n",
       "\n",
       "          ⎛                      _____________________________________________\n",
       "          ⎜                     ╱                                             \n",
       "    1     ⎜                    ╱  ⎛                                           \n",
       "───────── ⎜                   ╱   ⎜                                           \n",
       "α + β + γ ⎜                  ╱    ⎜                                           \n",
       "          ⎜             beta╱     ⎜               ⎛            -α - β - γ + 1 \n",
       "         ⋅⎝\\overline{q}⋅  ╲╱      ⎝\\overline{q}⋅α⋅⎝\\overline{q}              ⋅\n",
       "\n",
       "______________________________________________________________________________\n",
       "                   α                                                          \n",
       "             1    ⎞  ⎛                                                        \n",
       "         ─────────⎟  ⎜                                                    ────\n",
       "         α + β + γ⎟  ⎜                                                    α + \n",
       " γ  β  α⎞         ⎟  ⎜               ⎛            -α - β - γ + 1  γ  β  α⎞    \n",
       "m ⋅r ⋅w ⎠         ⎠ ⋅⎝\\overline{q}⋅γ⋅⎝\\overline{q}              ⋅m ⋅r ⋅w ⎠    \n",
       "\n",
       "________                                                                     _\n",
       "      γ                                                                     ╱ \n",
       "1    ⎞                                            1                        ╱  \n",
       "─────⎟                                        ─────────                   ╱   \n",
       "β + γ⎟                                        α + β + γ                  ╱    \n",
       "     ⎟   ⎛            -α - β - γ + 1  γ  β  α⎞                      beta╱     \n",
       "     ⎠  ⋅⎝\\overline{q}              ⋅m ⋅r ⋅w ⎠         ⋅(α + γ) + r⋅  ╲╱      \n",
       "\n",
       "_______________________________________________________________               \n",
       "                                    γ                        α                \n",
       "             ⎛                1    ⎞  ⎛                1    ⎞                 \n",
       "             ⎜            ─────────⎟  ⎜            ─────────⎟             ────\n",
       "             ⎜            α + β + γ⎟  ⎜            α + β + γ⎟             α + \n",
       "             ⎜  ⎛ α  β  γ⎞         ⎟  ⎜  ⎛ α  β  γ⎞         ⎟   ⎛ α  β  γ⎞    \n",
       "\\overline{q}⋅⎝m⋅⎝α ⋅β ⋅γ ⎠         ⎠ ⋅⎝w⋅⎝α ⋅β ⋅γ ⎠         ⎠  ⋅⎝α ⋅β ⋅γ ⎠    \n",
       "\n",
       "     ⎞\n",
       "     ⎟\n",
       "1    ⎟\n",
       "─────⎟\n",
       "β + γ⎟\n",
       "     ⎟\n",
       "     ⎠"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = (w*L + r*K + m*M).subs([(L,Lstar), (K,Kstar), (M,Mstar)])\n",
    "# simplify function does not work well here, but we can further simplify by hand\n",
    "C.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
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